Here we present a detailed numerical and analytical study of the transition of the failure process of
the fiber bundle model from the LLS to the ELS universality class when an initially
regular lattice of load transmitting connections is gradually randomized. Starting from a square
lattice we apply the Watts-Strogats rewiring technique [1]}, [2]} to
introduce long range random connections and study how the critical load and strain of the bundle, furthermore,
the statistics of the size of failure cascades change when the rewiring probability is gradually increased
at different degrees of disorder of the strength of nodes (fibers).
Our calculations revealed that the degree of strength disorder of nodes of the network has a substantial effect
on the transition. In particular, we show that the LLS-ELS transition sets on at a finite value of the rewiring
probability, which shifts to higher values as the degree of disorder is reduced. The transition
is limited to a well defined range of disorder, i.e. there exists a threshold disorder of nodes' strength
below which the randomization of the network structure does not provide any improvement neither of the
overall load bearing capacity nor of the cascade tolerance of the system. Based on the interplay of the
network structure and strength disorder we construct an analytical argument which provides a reasonable
description of the numerical findings.
